Abstract
Grid-tied inverters play a pivotal role in integrating distributed renewable energy sources into power systems. However, under weak grid conditions characterized by fluctuating grid impedance, background harmonics in grid voltage can severely distort the output current of inverters. Traditional capacitive voltage feedforward control strategies effectively suppress these harmonics but compromise system stability due to introduced feedforward paths. This paper proposes a weighted resonant capacitive voltage feedforward control strategy to enhance harmonic suppression while maintaining robust stability. By integrating a proportional-resonant (PR) controller into the feedforward path and optimizing weighting factors via a simulated annealing algorithm, the proposed method adaptively reshapes the system’s frequency response and output impedance. Simulation results validate the strategy’s effectiveness in improving power quality and stability under wide impedance variations.
1. Introduction
Grid-tied inverters are critical interfaces for renewable energy systems, enabling efficient power conversion and grid synchronization. However, weak grid conditions—marked by high grid impedance fluctuations—introduce significant background harmonics, destabilizing inverter operations. Existing solutions, such as capacitive voltage feedforward control, trade harmonic suppression for reduced stability margins. This work addresses this trade-off by proposing a hybrid control framework that combines resonant filtering, adaptive weighting, and impedance reshaping.
Key challenges include:
- Impedance Coupling: Interaction between inverter output impedance and grid impedance under weak grid conditions.
- Phase Lag: Digital control delays and feedforward-induced phase degradation.
- Harmonic Sensitivity: Noise interference in harmonic detection and compensation.
The proposed strategy leverages impedance analysis, PR-based frequency reshaping, and metaheuristic optimization to overcome these challenges.
2. System Modeling and Stability Analysis
2.1 Traditional Capacitive Voltage Feedforward Control
The grid-tied inverter system with an LCL filter is modeled as follows:
Main Circuit Parameters
| Parameter | Symbol | Value |
|---|---|---|
| Grid Voltage | UgUg | 311 V |
| DC Input Voltage | UdcUdc | 750 V |
| Filter Inductance (Inverter Side) | L1L1 | 5 mH |
| Filter Capacitance | CC | 15 µF |
| Filter Inductance (Grid Side) | L2L2 | 0.1 mH |
The control loop comprises a dual current feedback structure with PI regulators and delay compensation:Gi(s)=Kp+Kis,Gd(s)≈1−0.75sTs+0.083(1.5sTs)21+0.75sTs+0.083(1.5sTs)2Gi(s)=Kp+sKi,Gd(s)≈1+0.75sTs+0.083(1.5sTs)21−0.75sTs+0.083(1.5sTs)2
where TsTs is the sampling period.
The loop gain Tout(s)Tout(s) is derived as:Tout(s)=H1KPWMGd(s)Gi(s)s3L1L2C+s(L1+L2)Tout(s)=s3L1L2C+s(L1+L2)H1KPWMGd(s)Gi(s)
Stability analysis via Bode plots reveals that traditional feedforward reduces phase margin (PM) from 31.11° to 8.32°, highlighting instability risks.
2.2 Impedance Coupling Analysis
The grid-tied inverter and weak grid form a coupled system:ig(s)=Zo(s)Zo(s)+Zg(s)io(s)+Ug(s)Zo(s)+Zg(s)ig(s)=Zo(s)+Zg(s)Zo(s)io(s)+Zo(s)+Zg(s)Ug(s)
where Zo(s)Zo(s) and Zg(s)Zg(s) are the inverter output impedance and grid impedance, respectively. The Nyquist criterion requires:PMg=180∘−∠(Zg(j2πfw)Zo(j2πfw))>0PMg=180∘−∠(Zo(j2πfw)Zg(j2πfw))>0
Under weak grid conditions (Zg≠0Zg=0), impedance mismatches degrade stability.
3. Proposed Control Strategy
3.1 Resonant Capacitive Voltage Feedforward
A PR controller is integrated into the second-order feedforward path to reshape frequency characteristics:Gp(s)=Kp+2Krωcss2+2ωcs+ωo2Gp(s)=Kp+s2+2ωcs+ωo22Krωcs
The modified feedforward function becomes:Grf(s)=s2L2CKPWMGp(s)+1KPWMGrf(s)=KPWMs2L2CGp(s)+KPWM1
This enhances harmonic suppression at targeted frequencies (e.g., 3.7 kHz) while improving phase margins.
3.2 Adaptive Weighting via Simulated Annealing
To address impedance fluctuations, weighting factors λ1λ1 and λ2λ2 are introduced:Grf(s)=λ1s2L2CKPWMGp(s)+λ21KPWMGrf(s)=λ1KPWMs2L2CGp(s)+λ2KPWM1
A simulated annealing algorithm optimizes λ1λ1 and λ2λ2:
Simulated Annealing Workflow
- Initialize λ1=0.5λ1=0.5, λ2=0.5λ2=0.5, T0=100T0=100.
- Generate new solutions by perturbing λ1λ1 and λ2λ2.
- Evaluate cost function C(λ1,λ2)C(λ1,λ2), which penalizes low PM and high THD.
- Accept solutions probabilistically using:
P={1if ΔC<0,e−ΔC/Totherwise.P={1e−ΔC/Tif ΔC<0,otherwise.
- Iteratively reduce temperature TT until convergence.
Optimized Weighting Factors
| Grid Impedance LgLg | λ1λ1 | λ2λ2 |
|---|---|---|
| 0.2 mH | 0.3 | 0.2 |
| 0.8 mH | 0.2 | 0.1 |
| 1.3 mH | 0.1 | 0.5 |
4. Simulation Results
4.1 Harmonic Suppression Performance
Under balanced grid voltage with 3rd, 4th, and 5th harmonics (33 V each):
- Traditional Control: THD = 4.92%, unstable oscillations.
- Proposed Control: THD = 1.74–1.96%, stable operation.
4.2 Robustness to Impedance Fluctuations
For LgLg varying from 0.2 mH to 1.3 mH:
- Phase margin remains above 30° (vs. 8.32° for traditional methods).
- Current THD stays below 2% under nonlinear loads.
5. Conclusion
This paper presents a weighted resonant capacitive voltage feedforward control strategy for grid-tied inverters in weak grids. Key contributions include:
- Enhanced Stability: PR-based reshaping increases phase margin by 22° and gain margin by 7.3 dB.
- Adaptive Optimization: Simulated annealing dynamically adjusts weighting factors to maintain robustness against impedance fluctuations.
- Superior Harmonic Suppression: THD remains below 2% under severe grid distortions.
The proposed framework ensures grid-tied inverters achieve high power quality and stability, advancing renewable energy integration in weak grid environments.